Distribution of gaps between the inverses mod q

Let q be a positive integer, let I = I(q) and J = J(q) be subintervals of integers in [1, q] and let M be the set of elements of I that are invertible modulo q and whose inverses lie in J. We show that when q approaches infinity through a sequence of values such that φ(q)/q → 0, the r-spacing distribution between consecutive elements of M becomes exponential.